A Glimpse Into Discrete Differential Geometry

COMMUNICATION

A Glimpse into Discrete Differential Geometry

Keenan Crane and Max Wardetzky Communicated by Joel Hass

EDITOR’S NOTE. The organizers of the two-day AMS Short Course on Discrete Diﬀerential Geometry have kindly agreed to provide this introduction to the subject. The AMS Short Course runs in conjunction with the 2018 Joint Mathematics Meetings.

The emerging field of discrete differential geometry (DDG) studies discrete analogues of smooth geometric objects, providing an essential link between analytical descriptions and computation. In recent years it has unearthed a rich variety of new perspectives on applied problems in computational anatomy/biology, computational mechan- Figure 1. Discrete differential geometry reimagines ics, industrial design, computational architecture, and classical ideas from differential geometry without digital geometry processing at large. The basic philoso- reference to differential calculus. For instance, phy of discrete differential geometry is that a discrete surfaces parameterized by principal curvature lines object like a polyhedron is not merely an approximation are replaced by meshes made of circular of a smooth one, but rather a differential geometric object quadrilaterals (top left), the maximum principle in its own right. In contrast to traditional numerical anal- obeyed by harmonic functions is expressed via ysis which focuses on eliminating approximation error conditions on the geometry of a triangulation (top in the limit of refinement (e.g., by taking smaller and right), and complex-analytic functions can be smaller finite differences), DDG places an emphasis on replaced by so-called circle packings that preserve the so-called “mimetic” viewpoint, where key properties tangency relationships (bottom). These discrete of a system are preserved exactly, independent of how surrogates provide a bridge between geometry and large or small the elements of a mesh might be. Just computation, while at the same time preserving as algorithms for simulating mechanical systems might important structural properties and theorems. seek to exactly preserve physical invariants such as total energy or momentum, structure-preserving models of

Keenan Crane is assistant professor of computer science at Carnegie Mellon University. His e-mail address is kmcrane@cs discrete geometry seek to exactly preserve global geo- .cmu.edu. metric invariants such as total curvature. More broadly, Max Wardetzky is professor of mathematics at University of Göt- DDG focuses on the discretization of objects that do not tingen. His e-mail address is [email protected] naturally fall under the umbrella of traditional numerical .de. For permission to reprint this article, please contact: analysis. This article provides an overview of some of the [email protected]. themes in DDG. DOI: http://dx.doi.org/10.1090/noti1578

November 2017 Notices of the AMS 1153 COMMUNICATION

basic approach of DDG is to find alternative characterizations in the smooth setting that can be applied to discrete geometry in a natural way. With curvature, for instance, we can apply the fundamental theorem of calculus to Equation 1 to acquire a different statement: if 휑 is the angle from the horizontal line to the unit tangent 푇, then 푏 ∫ 휅 푑푠 = 휑(푏) − 휑(푎) mod 2휋. 푎 Put more simply: curvature is the rate at which the tangent Figure 2. A given geometric quantity from the smooth turns. This characterization can be applied naturally to setting, like curvature 휅, may have several reasonable our polygonal curve: along any edge the change in angle definitions in the discrete setting. Discrete is clearly zero. At a vertex it is simply the turning angle differential geometry seeks definitions that exactly 휃푖 ∶= 휑푖,푖+1 − 휑푖−1,푖 between the directions 휑푖−1,푖, 휑푖,푖+1 replicate properties of their smooth counterparts. of the two incident edges, yielding our first notion of discrete curvature: 퐴 (2) 휅푖 ∶= 휃푖 ∈ (−휋, 휋). The Game Are there other characterizations that also lead naturally Our article is organized around a “game” often played in to a discrete formulation? Yes: for instance we can discrete differential geometry in order to come up with a consider the motion of 훾 that most quickly reduces its discrete analogue of a given smooth object or theory: length. In the smooth case it is well known that the (1) Write down several equivalent definitions in the change in length with respect to a smooth variation smooth setting. 휂(푠) ∶ [0, 퐿] → ℝ2 that vanishes at the endpoints of the (2) Apply each smooth definition to an object in the curve is given by integration against curvature: discrete setting. 푑 퐿 (3) Analyze trade-offs among the resulting discrete | length(훾 + 휀휂) = − ∫ ⟨휂(푠), 휅(푠)푁(푠)⟩ 푑푠. definitions, which are invariably inequivalent. 푑휀 휀=0 0 Most often, none of the resulting discrete objects preserve Hence, the velocity that most quickly reduces length is all the properties of the original smooth one—a so- 휅푁. For a polygonal curve, we can simply differentiate 푛−1 called no free lunch scenario. Nonetheless, the properties the sum of the edge lengths 퐿 ∶= ∑푖=1 |훾푖+1 − 훾푖| with that are preserved often prove invaluable for particular respect to any vertex position. At a vertex 푖 we obtain applications and algorithms. Moreover, this activity yields 훾푖 − 훾푖−1 훾푖+1 − 훾푖 (3) 휕훾푖 퐿 = − =∶ 푇푖−1,푖 − 푇푖,푖+1, some beautiful and unexpected consequences—such as |훾푖 − 훾푖−1| |훾푖+1 − 훾푖| a connection between conformal geometry and pure i.e., just a difference of unit tangent vectors 푇푖,푖+1 along combinatorics, or a description of constant-curvature 2 consecutive edges. If 푁푖 ∈ ℝ is the unit angle bisector at surfaces that requires no definition of curvature! To vertex 푖, this difference can also be expressed as highlight some of the challenges and themes commonly 퐵 encountered in DDG, we first consider the simple example (4) 휅푖 푁푖 ∶= 2 sin(휃푖/2)푁푖, of the curvature of a plane curve.

Discrete Curvature of Planar Curves How do you deﬁne the curvature of a discrete curve? For a smooth arc-length parameterized curve 훾(푠) ∶ [0, 퐿] → ℝ2, curvature 휅 is classically expressed in terms of second 푑 derivatives. In particular, if 훾 has unit tangent 푇 ∶= 푑푠 훾 and unit normal 푁 (obtained by rotating 푇 a quarter turn in the counter-clockwise direction), then

푑2 푑 (1) 휅 ∶= ⟨푁, 푑푠2 훾⟩ = ⟨푁, 푑푠 푇⟩ . Suppose instead we have a polygonal curve with vertices 2 훾1, … , 훾푛 ∈ ℝ , as often used for numerical computation (see Figure 2, right). Here we hit upon the most elementary problem of discrete differential geometry: discrete geometric objects are often not sufficiently differentiable Figure 3. Different characterizations of curvature in (in the classical sense) for standard definitions to apply. the smooth setting naturally lead to different notions For instance, our curvature definition (Equation 1) causes of discrete curvature. (Here we abbreviate 푇푖,푖+1 and trouble, since at vertices our discrete curve is not twice 푇푖−1,푖 by 푢 and 푣, respectively.) differentiable, nor does it have well defined normals. The

1154 Notices of the AMS Volume 64, Number 10 COMMUNICATION providing a discretization of the curvature normal 휅푁. may end up with pointwise or integrated quantities in the A closely-related idea is to consider how the length of discrete case. a curve changes if we displace it by a small constant As one might imagine, there are many other possi- amount in the normal direction. As observed by Steiner, ble starting points for obtaining a discrete analogue of the new length of a smooth curve can be expressed as curvature. Eventually, however, all starting points end 퐿 up leading back to the same definitions, suggesting that (5) length(훾 + 휀푁) = length(훾) − 휀 ∫ 휅(푠) 푑푠. there may be only so many possibilities. For example, if 0 휙 ∶ ℝ2 → ℝ is the signed distance from a smooth closed Since this formula holds for any small piece of the curve 훾, then applying the Laplacian Δ yields the curva- curve, it can be used to obtain a notion of curvature ture of its level curves; in particular, Δ휙|휙=0 yields the at each point. How do we define normal offsets in the curvature of 훾. Likewise, if we apply the Laplacian to the polygonal case? At vertices we again encounter the issue signed distance function for a discrete curve, we recover that we have no notion of normals. One idea is to break the 휅퐴 on one side and 휅퐵 on the other. Yet another approach curve into individual edges which can then be translated is the theory of normal cycles (as discussed by Morvan), by 휀 along their respective normal directions. We can then related to the Steiner formula from Equation 5. Here, close the gaps between edges in a variety of ways: using rather than settle on a single normal 푁푖 at each vertex (A) a circular arc of radius 휀, (B) a straight line, or by we consider all unit vectors between the unit normals (C) extending the edges until they intersect (see Figure 3, of the two incident edges, ultimately leading back to the bottom left). If we then calculate the lengths for these first discrete curvature 휅퐴. The theory of normal cycles new curves, we get applies equally well to both smooth and polygonal curves, 푛−1 again reinforcing the perspective that the fundamental length퐴 = length(훾) − 휀 ∑푖=2 휃푖, 푛−1 behavior of geometry is neither inherently smooth nor length퐵 = length(훾) − 휀 ∑푖=2 2 sin(휃푖/2), 푛−1 discrete, but can be well captured in both settings by length = length(훾) − 휀 ∑ 2 tan(휃푖/2). 퐶 푖=2 picking the appropriate ansatz. More broadly, the fact Mirroring the observation in the smooth setting, we can that equivalent characterizations in the smooth setting now say that whatever change we observe in the length lead to different inequivalent definitions in the discrete provides a definition for discrete curvature. The first two setting is not special to the case of curves, but is one of are the same as ones we have seen already: the circular arc the central themes in discrete differential geometry. yielding the expression from Equation 2, and the straight From here, a natural question arises: which discrete line corresponding to Equation 4. The third one provides curvature is “best”? A traditional criterion for discrimi- yet another notion of discrete curvature nating among different discrete versions is the question 퐶 of convergence: if we consider finer and finer approxi- 휅푖 ∶= 2 tan(휃푖/2). mating polygons, will our discrete curvatures converge to Finally, in the smooth case it is also well known that the classical smooth one? However, convergence does not curvature has magnitude equal to the inverse of the always single out a best version: treated appropriately, all radius of the so-called osculating circle, which agrees with four of our discrete curvatures will converge. We must the curve up to second order. A natural way to define an osculating circle for a polygon is to take the circle passing through a vertex and its two neighbors. From the formula for the radius 푅푖 of a circumcircle in terms of the side lengths of the corresponding triangle, one easily gets a discrete curvature that is different from the ones we saw before: 퐷 (6) 휅푖 ∶= 1/푅푖 = 2 sin(휃푖)/푤푖, where 푤푖 ∶= |훾푖+1 − 훾푖−1|. Apart from merely being different The fundamental expressions, we can no- tice that 휅퐴, 휅퐵, and 휅퐶 behavior of are all invariant under geometry is a uniform scaling of Figure 4. Typically, not all properties of a smooth 퐷 the curve, whereas 휅 neither inherently object can be preserved exactly at the discrete level. scales like the smooth For curve-shortening flow, for example, 휅퐴 exactly curvature 휅. This sit- smooth nor preserves the total curvature, 휅퐵 exactly preserves uation demonstrates the center of mass, and with 휅퐷 the flow remains another common phe- discrete. stationary (up to rescaling) for any circular solution. nomenon in discrete However, no local definition of discrete curvature can differential geometry, namely that depending on which provide all three properties simultaneously. smooth characterization is used as a starting point, one

November 2017 Notices of the AMS 1155 COMMUNICATION therefore look beyond convergence, toward exact preservation of properties and relationships from the smooth setting. Which properties should we try to preserve? The answer of course depends on what we aim to use these curvatures for. As a toy example, consider the curve-shortening flow (depicted in Figure 4, top left), where a curve evolves according to the velocity that most quickly reduces its length. As discussed above, this velocity is equal to the curvature normal 휅푁. A smooth, simple curve evolving under this flow exhibits several basic properties: Figure 5. What is the simplicial analogue of a it has at all times total curvature 2휋, its center of conformal map? Requiring all angles to be preserved mass remains fixed, it tends toward a circle of vanishing is too rigid, forcing a global similarity (left). Asking only for preservation of so-called length cross ratios radius, and remains embedded for all time, i.e., no self- provides just the right amount of flexibility, crossings arise (Gage-Grayson-Hamilton). Do our discrete maintaining much of the structure found in the curvatures furnish these same properties? A numerical smooth setting such as invariance under Möbius experiment is shown in Figure 4. Here we evolve our transformations (right). polygon by a simple time-discrete flow 훾푖 ← 훾푖 + 휏휅푖푁푖 퐷 with a fixed time step 휏 > 0. For 휅 , 푁푖 is the unit vector along the circumradius; otherwise it is the unit angle bisector. Not surprisingly, 휅퐴 preserves total curvature Discrete Conformal Geometry (due to the fundamental theorem of calculus); 휅퐵 does A conformal map is, roughly speaking, a map that pre- not drift (consider summing Equation 3 over all vertices); serves angles (see Figure 1, bottom left). A good example and 휅퐷 has circular polygons as limit points (since is Mercator’s projection of the globe: even though area all velocities point toward the center of a common gets stretched out badly—making Greenland look much circle). However, no discrete curvature satisfies all three bigger than Australia!—the directions “north” and “east” properties simultaneously. Moreover, for a constant time remain at right angles, which is very helpful if you are step 휏 no such flow can guarantee that new crossings trying to navigate the sea. A beautiful fact about confor- do not occur. This situation illustrates the no free lunch mal maps is that any surface can be conformally mapped idea: no matter how hard we try, we cannot find a single to a space of constant curvature (“uniformization”), pro- discrete object that preserves all the properties of its viding it with a canonical geometry. This fact, plus the smooth counterpart. Instead, we have to pick and choose fact that conformal maps can be efficiently computed the properties best suited to the task at hand. (e.g., by solving sparse linear systems), have led in recent Suppose that instead of curvature flow, we consider years to widespread development of conformal mapping algorithms as a basic building block for digital geometry two other beautiful topics in the geometry of plane processing algorithms. In applications, discrete confor- curves: the Whitney–Graustein theorem, which classifies mal maps are used for everything from sensor network regular homotopy classes of curves by their total curva- layout to comparative analysis of medical or anatomical ture, and Kirchhoff’s famous analogy between motions data. Of course, to process real data one must be able to of a spherical pendulum and elastic curves, i.e., curves compute conformal maps on discrete geometry. that extremize the bending energy ∫퐿 휅2 푑푠 subject to 0 What does it mean for a discrete map to be conformal? boundary conditions. Among the curvatures discussed 퐴 As with curvature, one can play the game of enumerating above, only 휅 provides a discrete version of Whitney– several equivalent characterizations in the smooth setting. Graustein, but does not provide an exact discrete analogue 2 퐶 Consider for instance a map 푓 ∶ 푀 → 퐷 ⊂ ℂ from a disk- of Kirchhoff. Likewise, 휅 preserves the structure of the like surface 푀 with Riemannian metric 푔 to the unit disk Kirchhoff analogy, but not Whitney–Graustein. This kind 퐷2 in the complex plane. This map is conformal if it of no free lunch situation is a characteristic feature of preserves angles, if DDG. A similar obstacle is encountered in the theory of it preserves infinitesi- ordinary differential equations, where it is known that too few degrees of mal circles, if it can there are no numerical integrators for Hamiltonian sys- be expressed as a pair tems that simultaneously conserve energy, momentum, freedom relative to of real conjugate har- and the symplectic form. From a computational point of the number of monic functions 푓 = view, making judicious choices about which quantities 푎 + 푏푖, if it is a critical to preserve for which applications goes hand-in-hand constraints point of the Dirichlet 2 with providing formal guarantees on the reliability and energy ∫푀 |푑푓| 푑퐴, or robustness of algorithms. if it induces a new metric 푔̃ ∶= 푑푓 ⊗ 푑푓 that at each We now give a few glimpses into recent topics and point is a positive rescaling of the original one: 푔̃ = 푒2푢푔. trends in DDG. Each starting point leads down a path toward different

1156 Notices of the AMS Volume 64, Number 10 COMMUNICATION consequences in the discrete setting, and to algorithms relationship looks like a simple numerical approximation, with different computational tradeoffs. it turns out to provide a complete discrete theory that Oddly enough, the most elementary characterization preserves much of the structure found in the smooth of conformal maps, angle preservation, does not translate setting, with close ties to theories based on circles. An very well to the discrete setting (see Figure 5). Consider equivalent characterization is the preservation of length for instance a simplicial map that takes a triangulated cross ratios 픠푖푗푘푙 ∶= ℓ푖푗ℓ푘푙/ℓ푗푘ℓ푙푖 associated with each disk 퐾 = (푉, 퐸, 퐹) to a triangulation in the plane. Any edge ∈ 퐸; for a mesh embedded in ℝ푛 these ratios are map that preserves interior angles will be a similarity on invariant under Möbius transformations, again mimicking each triangle, i.e., it can only rigidly rotate and scale. But the smooth theory. This theory also leads to efficient, since adjacent triangles share edges, the scale factor for convex algorithms for discrete Ricci flow, which is a all triangles must be identical. Hence, the only discrete starting point for many applications in digital geometry surfaces that can be conformally flattened in this sense processing. are those that are (up to global scale) developable, i.e., More broadly, discrete conformal geometry and dis- that can be rigidly unfolded into the plane. This outcome crete complex analysis is an active area of research, with is in stark contrast to the smooth setting, where any elegant theories not only for triangulations but also for disk can be conformally flattened. This situation reflects lattice-based discretizations, which make contact with the a common scenario in DDG: rigidity, or what in finite topic of (discrete) integrable systems, discussed below. element analysis is sometimes called locking. There are Yet basic questions about properties like convergence, or simply too few degrees of freedom relative to the number descriptions that are compatible with extrinsic geometry, of constraints: we want to match angles at all 3퐹 corners, are still only starting to be understood. but have only 2푉 < 3퐹 degrees of freedom. Hence, if we insist on angle preservation we have no chance of Discrete Differential Operators capturing the flexibility of smooth conformal maps. Differential geometry and in particular Riemannian man- Other characterizations provide greater flexibility. One ifolds can be studied from many different perspectives. idea is to associate each vertex of our discrete disk 퐾 with In contrast to the purely geometric perspective (based a circle in the plane. A theorem of Koebe implies that one on, say, notions of distance or curvature), differential can always arrange these circles such that two circles are operators provide a very different point of view. One tangent if they belong to a shared edge and all boundary of the most fundamental operators in both physics and circles are tangent to a common circle bounding the rest. geometry is the Laplace–Beltrami operator Δ (or Laplacian For a regular triangular lattice approximating a region for short) acting on differential 푘-forms. It describes, for 푈 ⊂ ℂ, Thurston noticed that this map approximates a example, heat diffusion, wave propagation, and steady 2 smooth conformal map 푓 ∶ 푈 → 퐷 as the region is filled by state fluid flow, and is key to the Schrödinger equation smaller and smaller circles (see Figure 1, bottom), as later in quantum mechanics. It also provides a link between proved by Rodin and Sullivan. Unlike a traditional finite analytical and topological information: for instance, on element discretization, these so-called circle packings closed Riemannian manifolds the dimension of harmonic also preserve many of the basic structural properties of 푘-forms (i.e., those in the kernel of Δ) equals the di- conformal maps. For instance, composition with a Möbius mension of the 푘th cohomology—a purely topological transformation of the disk yields another uniformization quantity. The spectrum of the Laplacian (i.e., the list map, as in the smooth setting. More broadly, circle of eigenvalues) likewise reveals a great deal about the packings provide an unexpected bridge between geometry geometry of the manifold. For example, the first nonzero and combinatorics, since the geometry of a map is eigenvalue of the 0-form Laplacian provides an upper and 1 determined entirely by incidence relationships. On the a lower bound on optimally cutting a compact Riemannian flip side, this means a different theory is needed to manifold 푀 into two disjoint pieces of, loosely speaking, account for the geometry of irregular triangulations, as maximal volume and minimal perimeter (Cheeger-Buser). more commonly used in applications. These so-called Cheeger cuts have a wide range of ap- An alternative theory starts from the idea that under plications across machine learning and computer vision; a conformal map the Riemannian metric 푔 experiences a more broadly, eigenvalues and eigenfunctions of Δ help uniform scaling at each point: 푔̃ = 푒2푢푔. In other words, to generalize traditional Fourier-based simulation and vectors tangent to a given point 푝 ∈ 푀 shrink or grow signal processing to more general manifolds. by a positive factor 푒푢. In the simplicial setting 푔 is These observations motivate the study of discrete replaced by a piecewise Euclidean metric, i.e., a collection Laplacians, which can be defined even in the purely com- of positive edge lengths ℓ ∶ 퐸 → ℝ>0 that satisfy the binatorial setting of graphs. Here we briefly outline their triangle inequality in each face. Two such metrics ℓ, ℓ̃ definition for orientable finite simplicial 푛-manifolds, are then said to be discretely conformally equivalent if such as polyhedral surfaces, without boundary. Our ex- ̃ (푢푖+푢푗)/2 they are related by ℓ푖푗 = 푒 ℓ푖푗 for any collection of position is similar to what has become known as discrete discrete scale factors 푢 ∶ 푉 → ℝ. Though at first glance this exterior calculus. To this end, consider the simplicial boundary operators 휕푘 ∶ 퐶푘 → 퐶푘−1 acting on 푘-chains 1See “Circle Packing” in the December 2003 Notices www.ams (i.e., formal linear combinations of 푘-simplices). The cor- 푘 .org/notices/200311/fea-stephenson.pdf. responding dual spaces (cochains) 퐶 ∶= Hom(퐶푘, ℝ) and

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푘 푘+1 2 3 respective dual operators 훿푘 ∶ 퐶 → 퐶 give rise to the surface 푓 ∶ 푈 ⊂ ℝ → ℝ where the directional derivatives chain complex 푓푢 and 푓푣 along the coordinate directions satisfy |푓푢|푣 = {0} → 퐶0 → 퐶1 → … → 퐶푛 → {0}. |푓푣|푢 = 0, i.e., partial derivatives with respect to one parameter have constant length along the parameter The chain property says that 훿푘 ∘ 훿푘−1 = 0, and one hence lines of the other parameter. The special case of rhombic obtains simplicial cohomology 퐻푘 ∶= ker(훿 )/im(훿 ). 푘 푘−1 tilings (|푓푢| = |푓푣| = 1) are known as (strong) Chebyshev 푘 To define a Laplacian in this setting, we equip each 퐶 nets. Can every smooth surface be wrapped in a stocking? ∗ with a positive definite inner product (⋅, ⋅)푘, and let 훿푘 be Locally (i.e., in a small patch around any given point) the adjoint operator with respect to these inner products, the answer is “yes.” Globally, however, there are severe ∗ i.e., (훿푘훼, 훽)푘+1 = (훼, 훿푘 훽)푘 for all 훼, 훽. The Laplacian on obstructions to doing so, which provide some fascinating 푘-cochains is then defined as connections to physics. ∗ ∗ Δ푘 ∶= 훿푘 훿푘 + 훿푘−1훿푘−1. Consider for instance the special case of so-called K-surfaces, characterized by constant Gauß curvature The resulting space of harmonic 푘-cochains, {훼 ∈ 퐾 = −1. Every K-surface admits a parameterization 푓 ∶ 퐶푘|Δ 훼 = 0} is then isomorphic to 퐻푘—just as in 푘 푈 ⊂ ℝ2 → ℝ3 aligned with the two transversal asymptotic the smooth setting. This fact is independent of the directions along which normal curvature vanishes. Hence, choice of inner product, mirroring the fact that coho- if 푁 is the unit surface normal then mology depends only on topological structure. Likewise, for any inner product one obtains a discrete Hodge (7) ⟨푓푢푢, 푁⟩ = ⟨푓푣푣, 푁⟩ = 0. decomposition Asymptotic parameterizations are weak Chebyshev nets 푘 ∗ 퐶 = ker(Δ푘) ⊕ im(훿푘−1) ⊕ im(훿푘 ), since where here the subspaces do depend on the choice of (8) 푎 ∶= |푓푢|, 푏 ∶= |푓푣| satisfy 푎푣 = 푏푢 = 0. inner product. Moreover, one can show that the angle 휙 between At this point we return again to the game of DDG: asymptotic lines satisfies the sine-Gordon equation which choice of inner product is best? A trivial inner product leads to purely combinatorial graph Laplacians, (9) 휙푢푣 − 푎푏 sin 휙 = 0, which do not (in general) converge to their smooth coun- and conversely, every solution to the sine-Gordon equa- terparts (e.g., when approximating a smooth manifold by tion describes a parameterized K-surface. Hilbert used a polyhedral one). Another choice is to consider linear this equation (and Chebyshev nets) to prove that the com- interpolation of 푘-cochains over 푛-dimensional simplices, plete hyperbolic plane cannot be embedded isometrically resulting in what is known as Whitney elements. For 푛 = 2, into ℝ3. More generally, the sine-Gordon equation has we get the so-called cotan Laplacian (Pinkall and Polth- attracted much interest both in mathematics as an exam- ier), which is widely used in digital geometry processing. ple of an infinite-dimensional integrable system, and in Though other choices are possible, we again encounter physics as an example of a system that admits remarkably a no free lunch situation: no choice of inner product stable soliton solutions, akin to waves that travel uninter- can preserve all the properties of the smooth Laplacian. rupted all the way across the ocean. Another key property Which properties do we care about? Beyond convergence, of the sine-Gordon equation is the existence of a so-called perhaps the most desirable properties are the maximum spectral parameter 휆 > 0: Equation 9 is invariant under principle (which ensures, for instance, proper behavior a rescaling 푎 → 휆푎 and 푏 → 휆−1푏, giving rise to a one- for heat flow), and the property that, for flat domains, parameter associated family of K-surfaces. Geometrically, linear functions are in the kernel (leading to a proper the parameter 휆 rescales the edges of parallelograms definition of barycentric coordinates). For general un- while preserving the angle between asymptotic lines. structured meshes there are no discrete Laplacians with Do these properties depend critically on the smooth all of these properties. However, certain types of meshes nature of the solutions, or can they also be faithfully (such as weighed Delaunay triangulations) do indeed allow captured in the discrete setting? Hirota derived such a for “perfect” discrete Laplacians, offering a connection discrete version without any reference to geometry. Later between geometry and (discrete) differential operators. Bobenko and Pinkall suggested a geometric definition of discrete K-surfaces that recovers Hirota’s equation. In Discrete Integrable Systems their setting, discrete K-surfaces are defined as discrete Another topic that has provided inspiration for many (weak) Chebyshev nets with the additional property that ideas in DDG is parameterized surface theory. Consider all four edges incident to any vertex lie in a common for instance the problem of dressing a given surface plane. The last requirement is a natural discrete analogue by a fishnet stocking, i.e., a woven material composed of of Equation 7. This definition of discrete K-surfaces also inextensible yarns following transversal “warp” and “weft” comes with a spectral parameter 휆 and results in Hirota’s directions (see Figure 6, right). This task corresponds to discrete sine-Gordon equation—without requiring any decorating a surface with a tiling where each vertex notion of discrete Gauß curvature. Only recently has a is incident to four parallelograms. Infinitesimally, such a discrete version of Gauß curvature been suggested that tiling is known as a weak Chebyshev net (Chebyshev 1878), results in discrete K-surfaces indeed having constant and locally corresponds to a regularly parameterized negative Gauß curvature.

1158 Notices of the AMS Volume 64, Number 10 COMMUNICATION

Image Credits Figures 1–6 courtesy of Keenan Crane and Max Wardetzky. Figure 7 courtesy of B. Schneider. Author photo of Keenan Crane courtesy of Keenan Crane. Author photo of Max Wardetzky courtesy of Max Wardetzky.

Figure 6. Left: two discrete parameterizations of a pseudosphere (constant Gauß curvature 퐾 = −1), one ABOUT THE AUTHORS with a Chebychev net along asymptotic directions Keenan Crane works at the inter- (left) and another along principal curvature lines face between diﬀerential geometry (right). Right: a discrete Chebyshev net on a surface and geometric algorithms. His sci- of varying curvature, resembling the weft and warp entiﬁc career started out studying directions of a woven material. Pluto—back when it was still a planet!

Keenan Crane

Max Wardetzky’s interest in discrete differential geometry comes from his outstanding and truly inspiring teachers in differential geometry and his exposure to marvelous people in computer graphics. Figure 7. Discrete parameterized surfaces play a role in architectural geometry, where special incidence Max Wardetzky relationships on quadrilaterals translate to manufacturing constraints like zero nodal torsion, or offset surfaces of constant thickness. Here a curvature line parameterized surface discretized by a conical net is used in the design of a railway station.

For discrete K-surfaces with all equal edge lengths (i.e., the rhombic case) the four neighboring vertices of a given vertex must lie on a common circle. By considering a subset of the diagonals of the quadrilaterals, one obtains another quad mesh with the property that all quads have a circumscribed circle, resulting in so- called cK-nets (see Figure 6, left). In the discrete setting, regular networks of circular quadrilaterals play the role of curvature line parameterized surfaces (as in Figure 1, top left). This transformation therefore mimics the smooth setting, where the angle bisectors of asymptotic lines are lines of principal curvature. More broadly, the theory of quad nets with special incidence relationships is closely linked to physical manufacturing considerations in the field of architectural geometry. For example, a quad net is conical if the four quads around each vertex are tangent to a common cone—such surfaces admit face offsets of constant width, making them attractive for the construction of (for instance) glass-paneled structures, as in Figure 7. For further reading, see Discrete Differential Geometry (2008, Alexander Bobenko ed.).

November 2017 Notices of the AMS 1159